Copulas and spectral measures: Statistical modelling under long-range dependence

许多观察到的现象不能用线性相关结构来解释。此外，经常观察到的时间序列表现出线性或非线性的长程相关性。Copula函数提供了一种建模依赖性的一般方法。尽管有大量的文献，但目前的Copula模型往往不容易解释，同时也无法捕捉到极端情况的依赖结构。此外，在这个阶段，似乎没有结果copula在长期依赖的背景下。这些问题在本项目中得到解决。该项目分为三个主要部分。在第一部分中，应用方面的统计建模与copula和极值copula下独立或弱依赖进行了探讨。我们专注于寻找模型捕捉广泛的依赖结构的多元极端。在第二部分和第三部分的统计推断（极值）Copula和其他模型参数的线性（第二部分）和非线性（第三部分）的有限或无限的二阶矩的长记忆过程的背景下解决。将考虑估计、检验（拟合优度、线性）和条件风险测量。由于copula结构和长程相关性，非标准极限定理预计将出现在第二部分和第三部分。

Many observed phenomena cannot be explained by linear dependence structures. Moreover,often observed time series exhibit either linear or nonlinear long-range dependence. A general approach to modelling dependence is provided by copulas. In spite of an extended literature, current copula models are often not easy to interpret and at the same time fail to capture the dependence structure of extremes. Moreover, at this stage, no results on copulas in the context of long-range dependence seem to be available. These issues are addressed in this project. The project is divided into three main parts. In Part I, applied aspects of statistical modelling with copulas and extremal copulas are explored under independence or weak dependence. We focus on finding models capturing a wide range of dependence structures for multivariate extremes. In Parts II and III statistical inference for (extreme value) copulas and other model parameters is addressed in the context of linear (Part II) and nonlinear (Part III) long-memory processes with finite or infinite second moments. Estimation, testing (goodness-of-fit, linearity) and conditional risk measures will be considered. Due to the copula structure and long-range dependence, nonstandard limit theorems are expected to emerge in Parts II and III.